Vertex connected edge highlighting
In this case the vertice wasn't highlighted as expected because the Mirror modifier, which was present in stack, had button Adjust Edit cage to modifier's result turned on.



This means the result of modifier's work was visible while in Edit mode and editable more or less (so you could select verts mirrored by it, for example). 

But in this case part of mesh doesn't exist (because result isn't applied yet) and hence existing vertices of the mesh aren't highlighted as usual (when edges which are connected by these vertices are highlighted too) rather than one by one. Edge is being highlighted only when 2 vertices were selected.



Note: vertices' size is increased for demonstration purposes.

To work without this behaviour uncheck Adjust Edit cage to modifier's result in order to directly interact only with real vertices of the mesh. You can check it if needed to preview changes when it's important for the workflow.

Read more about interface of modifiers and meaning of the present buttons.